ITPEC IP October 2018 Question 2

Topic: Basic TheoryAlso asked in Spring 2021

Which of the following is obtained by multiplying the binary value 10110 by three (3)?

Explanation

1000010 — multiplying by 3 is the same as adding the number shifted left once (×2) to itself (×1).

  • •3x = 2x + x: 10110 shifted left one place is 101100
  • •101100 + 10110 = 1000010, with carries
  • •Check in decimal: 10110 is 16 + 4 + 2 = 22, and 22 × 3 = 66 = 64 + 2, which is 1000010

Answer (c)

Why not others:

  • •(a) 111010 — adds 101100 and 10110 digit by digit but drops every carry (an XOR); it is 58
  • •(b) 111110 — writes a 1 wherever either number has a 1 (an OR), again ignoring carries; it is 62
  • •(d) 10110000 — shifts left three places, which multiplies by 23 = 8, not by 3; it is 176

Key rule: A left shift by n multiplies by 2n; to multiply by a number that is not a power of 2, add shifted copies (3x = 2x + x) and carry as in decimal addition.

Basic Theory · AI-written, may contain errors

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